I’ve connected 1 ESC to the Navio2 (channel 1) with all the 3 wires. There is a servo on the channel 7. The issue is, that servo is not doing anything, only if I connect it directly to the X8R receiver’s channel 7.

It turned out that: DYS HV40A ESC seems to be OPTO, so it does not have built-in BEC.

Yes you will need to use the 5V one as that is the standard voltage for servos. However since you are using 6S lipo it might be better to use this voltage regulator instead since it supports up to 30V input, whereas the ones I mentioned before only support up to 23V.